Convergence of Collatz Sequences: Procedure to Prove the Collatz Conjecture

TL;DR

This paper proposes a sequential proof method for the Collatz conjecture by analyzing sequence convergence patterns, utilizing pre-proven data, and demonstrating near-complete computational verification.

Contribution

It introduces a novel procedure based on pattern analysis and pre-proven sequences to systematically prove the convergence of all Collatz sequences.

Findings

Over 99% of sequences verified to converge

Pattern analysis reveals quartets and octets in sequences

Computational process nearing completion for full proof

Abstract

As Collatz conjecture is still to be proved, a method to arrive at the complete proof is explored here. Conceptually, the process relies on the pre-proven sequence data and the method follows the confirmation of the convergence of the Collatz sequence for all the natural numbers in a sequential forward manner (ascending order of n). Hence, all the numbers less than n have been proved to converge before beginning to prove the convergence of the sequence for the number n. Initially, the the nature of problem was explored and explained the reason for some of the remarkable properties. Then, the pattern analysis of a first few sequences data led to the concept of pre-proven sequence. Then, the origin of quartets and subsequently, the octet patterns are described. The "Even-Odd" combinations and the concept of (8*v + c) gave a clear picture for the overall process of evaluation for the convergence. Then onwards, individual set of numbers are analyzed and the convergence confirmations are arrived at. The process of computing involves the concept of combing. Finally, the compilation of all the sets together would give the completion of the proof. Here, the procedure for the computation is presented with example and the current status of computation ($>$99\% completed) while the laborious computation is in progress (for the remaining $<$ 1\%). It is expected that the length of the PSO number list is finite and all the numbers would converge.